Fabric 1.20.1
Better End Cities Villa 1.20.1
BetterEnd 4.0.11
Bigger Better End Cities 1.20.1-1.20.4

When I loot an elytra on an item frame in an End Ship, Lootr does not update it at all. No elytra gets regenerated for other players who haven't looted the same item frame.

It appears that Bigger Better End Cities may be the issue as it has its own handleMetadata which is potentially being used instead of the Vanilla.

Unfortunately I don't really see a way to resolve this without manually mixing in to the Bigger End Cities mod itself.

Lootr's Elytra functionality replaces the item frame with a chest beneath it.

If you are seeing an item frame, then it means that the world was either generated before Lootr was installed, or the Elytra conversion option was disabled when the structure was generated.

My recommendation would be to use WorldEdit or MCA Selector to specifically regenerate the chunk the end ship is contained in while Lootr is enabled and the Elytra conversion option is enabled.